In today's world, yes it is cocky for British people to refer to England as Great Britain. They are no longer an "Empire" , just an island. They are not even the economically strongest country (Germany). England is no longer great, just an average European country. Though they have strong patriotism, which I would like to see more of in the USA.
We are called "Great Britain", don't like it then you can piss right off. It is our right to call our country what we wish and if we wish to be called "Great" Britain, then we will be bloody well called Great Britain. And in any case, England is not Britain, England is part of Britain.
Great Britain has been called Great Britain for many years. Britain is made up of England, Northern Ireland, Scotland and Wales - not just England! Being a Brit, I do not think of my country as superior to others and when calling my country 'Great Britain' I do not think that the 'great' makes the country better than others. It is called known as the United Kingdom anyway.
It isn't a bit cocky for Britons to call their country 'Great' Britain, because of the historical context behind the usage of the name. Because Great Britain contains Scotland, Ireland, and Wales, it can be labeled in this context without fear of being cocky. This could only be taken as cocky if they were referring to the country in a different manner.
The word "Great" in "Great Britain" derives from an older name for the area. Centuries ago, writers called it Britannia Major ("Great Britain") to distinguish it from another region that was called Britannia Minor ("Lesser Britain"). Thus, "Great" in this case is not an expression of patriotic pomp but refers to a geographical aspect of the country.